rtree-multi

In-memory immutable R-tree implementation for n dimensions.

Status: in development

An R-tree is a commonly used spatial index.

This was fun to make, has an elegant concise algorithm, is thread-safe, fast, and reasonably memory efficient (uses structural sharing).

The algorithm to achieve immutability is cute. For insertion/deletion it involves recursion down to the required leaf node then recursion back up to replace the parent nodes up to the root. The guts of it is in Leaf.java and NonLeaf.java.

Iterator support requires a bookmark to be kept for a position in the tree and returned to later to continue traversal. An immutable stack containing the node and child index of the path nodes came to the rescue here and recursion was abandoned in favour of looping to prevent stack overflow (unfortunately java doesn't support tail recursion!).

Maven site reports are here including javadoc.

Features

• supports n dimensions
• immutable R-tree suitable for concurrency
• Guttman's heuristics (Quadratic splitter) (paper)
• R*-tree heuristics (paper)
• Customizable splitter and selector
• search is O(log(n)) on average
• insert, delete are O(n) worst case
• balanced delete
• uses structural sharing
• JMH benchmarks
• visualizer included
• decent unit test code coverage
• R*-tree performs ? searches/second returning 22 entries from a tree of 38,377 Greek earthquake locations on i7-920@2.67Ghz (maxChildren=4, minChildren=1). Insert at ? entries per second.
• requires java 1.8 or later

2 dimensions

Number of points = 1000, max children per node 8:

Quadratic split	R*-tree split

Notice that there is little overlap in the R*-tree split compared to the Quadratic split. This should provide better search performance (and in general benchmarks show this).

3 dimensions

Given the 38,377 data points of greek earthquakes (lat, long, time) from 1964 to 2000, the data is scanned to establish the ranges for each coordinate then normalized to a [0,1] range. The points are shuffled then added to an R-tree with minChildren=2 and maxChildren=4 using either the R* heuristics or standard R-tree heuristics. Visualization of the bounding boxes at nodes by method and depth is below.

The plots below are generated from the same shuffle of data points.

Quadratic split	R*-tree split

Getting started

Add this maven dependency to your pom.xml:

<dependency>
  <groupId>com.github.davidmoten</groupId>
  <artifactId>rtree-multi</artifactId>
  <version>VERSION_HERE</version>
</dependency>

Instantiate an R-Tree

Use the static builder methods on the RTree class:

// create an R-tree using Quadratic split with max
// children per node 4, min children 2 (the threshold
// at which members are redistributed)
RTree<String, Geometry> tree = RTree.create();

You can specify a few parameters to the builder, including minChildren, maxChildren, splitter, selector:

RTree<String, Geometry> tree = RTree.minChildren(3).maxChildren(6).create();

Geometries

The following geometries are supported for insertion in an RTree:

• Rectangle
• Point

Generic typing

If for instance you know that the entry geometry is always Point then create an RTree specifying that generic type to gain more type safety:

RTree<String, Point> tree = RTree.create();

R*-tree

If you'd like an R*-tree (which uses a topological splitter on minimal margin, overlap area and area and a selector combination of minimal area increase, minimal overlap, and area):

RTree<String, Geometry> tree = RTree.star().maxChildren(6).create();

See benchmarks below for some of the performance differences.

Add items to the R-tree

When you add an item to the R-tree you need to provide a geometry that represents the 2D physical location or extension of the item. You can use these factory methods to create your geometries:

• Rectangle.create
• Point.create

To add an item to an R-tree:

RTree<T,Geometry> tree = RTree.dimensions(3).create();
tree = tree.add(item, Point.create(10, 20, 30));

or

tree = tree.add(Entry.entry(item, Point.create(10, 20, 30));

Important note: being an immutable data structure, calling tree.add(item, geometry) does nothing to tree, it returns a new RTree containing the addition. Make sure you use the result of the add!

Remove an item in the R-tree

To remove an item from an R-tree, you need to match the item and its geometry:

tree = tree.delete(item, Point.create(10, 20, 30));

or

tree = tree.delete(entry);

Important note: being an immutable data structure, calling tree.delete(item, geometry) does nothing to tree, it returns a new RTree without the deleted item. Make sure you use the result of the delete!

Custom geometries

You can also write your own implementation of Geometry. An implementation of Geometry needs to specify methods to:

• check intersection with a rectangle (you can reuse the distance method here if you want but it might affect performance)
• provide a minimum bounding rectangle
• implement equals and hashCode for consistent equality checking
• measure distance to a rectangle (0 means they intersect). Note that this method is only used for search within a distance so implementing this method is optional. If you don't want to implement this method just throw a RuntimeException.

For the R-tree to be well-behaved, the distance function if implemented needs to satisfy these properties:

• distance(r) >= 0 for all rectangles r
• if rectangle r1 contains r2 then distance(r1)<=distance(r2)
• distance(r) = 0 if and only if the geometry intersects the rectangle r

Searching

The advantage of an R-tree is the ability to search for items in a region reasonably quickly. On average search is O(log(n)) but worst case is O(n).

Search methods return Iterable sequences:

Iterable<Entry<T, Geometry>> results =
    tree.search(Rectangle.create(0, 0, 2, 2));

or search for items within a distance from the given geometry:

Iterable<Entry<T, Geometry>> results =
    tree.search(Rectangle.create(0, 0, 2, 2), 5.0);

To return all entries from an R-tree:

Iterable<Entry<T, Geometry>> results = tree.entries();

Search with a custom geometry

Suppose you make a custom geometry like Polygon and you want to search an RTree<String, Point> for points inside the polygon. This is how you do it:

RTree<String, Point> tree = RTree.dimensions(2).create();
Func2<Point, Polygon, Boolean> pointInPolygon = ...
Polygon polygon = ...
...
entries = tree.search(polygon, pointInPolygon);

The key is that you need to supply the intersects function (pointInPolygon) to the search. It is on you to implement that for all types of geometry present in the RTree. This is one reason that the generic Geometry type was added in rtree 0.5 (so the type system could tell you what geometry types you needed to calculate intersection for) .

Search with a custom geometry and maxDistance

As per the example above to do a proximity search you need to specify how to calculate distance between the geometry you are searching and the entry geometries:

RTree<String, Point> tree = RTree.create();
Func2<Point, Polygon, Boolean> distancePointToPolygon = ...
Polygon polygon = ...
...
entries = tree.search(polygon, 10, distancePointToPolygon);

Example

import com.github.davidmoten.rtree-multi.RTree;

RTree<String, Point> tree = RTree.maxChildren(5).create();
tree = tree.add("DAVE", Point.create(10, 20))
           .add("FRED", Point.create(12, 25))
           .add("MARY", Point.create(97, 125));
 
Iterable<Entry<String, Point>> entries =
    tree.search(Rectangle.create(8, 15, 30, 35));

How to configure the R-tree for best performance

Check out the benchmarks below, but I recommend you do your own benchmarks because every data set will behave differently. If you don't want to benchmark then use the defaults. General rules based on the benchmarks:

• for data sets of <10,000 entries use the default R-tree (quadratic splitter with maxChildren=4)
• for data sets of >=10,000 entries use the star R-tree (R*-tree heuristics with maxChildren=4 by default)

Watch out though, the benchmark data sets had quite specific characteristics. The 1000 entry dataset was randomly generated (so is more or less uniformly distributed) and the Greek dataset was earthquake data with its own clustering characteristics.

What about memory use?

To minimize memory use you can use geometries that store single precision decimal values (float) instead of double precision (double). Here are examples:

// create geometry using double precision 
Rectangle r = Rectangle.create(1.0, 2.0, 3.0, 4.0);

// create geometry using single precision
Rectangle r = Rectangle.create(1.0f, 2.0f, 3.0f, 4.0f);

Visualizer

To visualize a 2D R-tree (only) in a PNG file of size 600 by 600 pixels just call:

tree.visualize(600,600)
    .save("target/mytree.png");

The result is like the images in the Features section above.

Visualize as text

The RTree.asString() method returns output like this:

mbr=Rectangle [mins=[10.0, 4.0], maxes=[62.0, 85.0]]
  mbr=Rectangle [mins=[28.0, 4.0], maxes=[34.0, 85.0]]
    entry=Entry [value=2, geometry=Point [29.0, 4.0]]
    entry=Entry [value=1, geometry=Point [28.0, 19.0]]
    entry=Entry [value=4, geometry=Point [34.0, 85.0]]
  mbr=Rectangle [mins=[10.0, 45.0], maxes=[62.0, 63.0]]
    entry=Entry [value=5, geometry=Point [62.0, 45.0]]
    entry=Entry [value=3, geometry=Point [10.0, 63.0]]

Serialization

Serialization is not supported directly in rtree-multi. Your best bet is to serialize entries from and RTree as you like (just the entries) and then use bulk loading when deserializing. Bulk loading is performed like this:

// deserialize the entries from disk (for example)
List<Entry<Thing, Point> entries = ...
// bulk load
RTree<Thing, Point> tree = RTree.maxChildren(28).star().create(entries); 

How to build

git clone https://github.com/davidmoten/rtree-multi.git
cd rtree-multi
mvn clean install

How to run benchmarks

Benchmarks are provided by

mvn clean install -P benchmark

Notes

The Greek data referred to in the benchmarks is a collection of some 38,377 entries corresponding to the epicentres of earthquakes in Greece between 1964 and 2000. This data set is used by multiple studies on R-trees as a test case.

Results

These were run on i7-920 @2.67GHz with rtree version 0.9-RC1:

# JMH version: 1.21
# VM version: JDK 1.8.0_201, Java HotSpot(TM) 64-Bit Server VM, 25.201-b09
# VM invoker: /usr/lib/jvm/java-8-oracle/jre/bin/java
# VM options: -Xmx512m
# Warmup: 3 iterations, 10 s each
# Measurement: 10 iterations, 10 s each
# Timeout: 10 min per iteration
# Threads: 1 thread, will synchronize iterations
# Benchmark mode: Throughput, ops/time

Benchmark                                                        Mode  Cnt        Score       Error  Units
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren004       thrpt   10  1021844.129 ± 16977.542  ops/s
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren010       thrpt   10   931789.301 ± 33477.764  ops/s
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren032       thrpt   10   304816.886 ±  5674.364  ops/s
BenchmarksRTree.rStarTreeSearchOf1000PointsMaxChildren128       thrpt   10   606188.199 ±  7553.533  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren004  thrpt   10   527595.512 ± 34400.389  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren010  thrpt   10   194793.663 ±  5385.855  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren032  thrpt   10   297597.166 ± 40175.027  ops/s
BenchmarksRTree.rStarTreeSearchOfGreekDataPointsMaxChildren128  thrpt   10   221605.832 ±  9332.555  ops/s
BenchmarksRTree.searchNearestGreek                              thrpt   10     3534.733 ±   177.370  ops/s